| Summary: | The purpose of the thesis is to analyse the management of various forms of risk that affect entire insurance portfolios and thus cannot be eliminated by increasing the number of policies, like catastrophes, financial market events and fluctuating insurance risk conditions. Three distinct frameworks are employed. First, we study the optimal design of a catastrophe-related index that an insurance company may use to hedge against catastrophe losses in the incomplete market. The optimality is understood in terms of minimising the remaining risk as proposed by Follmer and Schweizer. We compare seven hypothetical indices for an insurance industry comprising several companies and obtain a number of qualitative and formula-based results in a doubly stochastic Poisson model with the intensity of the shot-noise type. Second, with a view to the emergence of mortality bonds in life insurance and longevity bonds in pensions, the design of a mortality-related derivative is discussed in a Markov chain environment. We consider longevity in a scenario where specific causes of death are eliminated at random times due to advances in medical science. It is shown that bonds with payoff related to the individual causes of death are superior to bonds based on broad mortality indices, and in the presence of only one cause-specific derivative its design does not affect the hedging error. For one particular mortality bond linked to two causes of death, we calculate the hedging error and study its dependence on the design of the bond. Finally, we study Pareto-optimal risk exchanges between a group of insurance companies. The existing one-period theory is extended to the multiperiod and continuous cases. The main result is that every multiperiod or continuous Pareto-optimal risk exchange can be reduced to the one-period case, and can be constructed by pre-setting the ratios of the marginal utilities between the group members.
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